MHD stagnation point flow of Casson fluid over a convective stretching sheet considering thermal radiation,slip condition,and viscous dissipation

This study presents the problem of MHD stagnation point flow of Casson fluid over a convective stretching sheet considering thermal radiation, slip condition, and viscous dissipation. The partial differential equations with the corresponding boundary conditions that govern the fluid flow are reduced to a system of highly nonlinear ordinary differential equations using scaling group transformations. The fourth-order method along shooting technique is applied to solve this system of boundary value problems numerically. The effects of flow parameters on the velocity, temperature, and concentration profiles are presented via graphs. The impact of the physical parameters on the skin friction coefficient reduced Nusselt numbers and reduced Sherwood numbers are investigated through tables. Comparison of the present findings with the previously published results in the literature shows an excellent agreement. It is also noted that a rise in the Eckert number results in a drop in the temperature of the fluid in the thermal boundary layer region of the fluid flow.     

Thermal radiation and diffusion effects in MHD Williamson and Casson fluid flows past a slendering stretching surface

The article is presented to analyze the magnetohydrodynamic Casson and Williamson fluids flow over a stretched surface of variable thickness by including the conditions of thermal radiation, velocity slip, temperature, and concentration slip. The equations governing the flow characteristics are transformed to ordinary differential equations by applying similarity transformations. The solution of the simplified equations is obtained by the numerical bvp5c Matlab package. The behavior for Williamson and Casson fluid cases is explored and discussed with the impact of sundry parameters on the flowing fluid, thermal, and diffusion fields. The profiles under the impact of parameters are depicted through graphs. Also, we evaluated the performance of local Nusselt and Sherwood numbers along with the friction of the wall and are displayed through tables. We found that the temperature and mass transfer distribution is low in Williamson fluid when compared to Casson fluid flow. The computed results indicate that the flow, thermal and concentration boundary layer characteristics of Williamson and Casson fluids are not unique.     

Entropy analysis of unsteady MHD three-dimensional flow of Williamson nanofluid over a convectively heated stretching sheet

This article addresses an investigation of the entropy analysis of Williamson nanofluid flow in the presence of gyrotactic microorganisms by considering variable viscosity and thermal conductivity over a convectively heated bidirectionally stretchable surface. Heat and mass transfer phenomena have been incorporated by taking into account the thermal radiation, heat source or sink, viscous dissipation, Brownian motion, and thermophoretic effects. The representing equations are nonlinear coupled partial differential equations and these equations are shaped into a set of ordinary differential equations via a suitable similarity transformation. The arising set of ordinary differential equations was then worked out by adopting a well-known scheme, namely the shooting method along with the Runge-Kutta-Felberge integration technique. The effects of flow and heat transfer controlling parameters on the solution variables are depicted and analyzed through the graphical presentation. The survey finds that magnifying viscosity parameter, Weissenberg number representing the non-Newtonian Williamson parameter cause to retard the velocity field in both the directions and thermal conductivity parameter causes to reduce fluid temperature. The study also recognizes that enhancing magnetic parameters and thermal conductivity parameters slow down the heat transfer rate. The entropy production of the system is estimated through the Bejan number. It is noticeable that the Bejan number is eminently dependent on the heat generation parameter, thermal radiation parameter, viscosity parameter, thermal conductivity parameter, and Biot number. The skillful accomplishment of the present heat and mass transfer system is achieved through the exteriorized choice of the pertinent parameters.     

Mathematical modeling and study of MHD flow of Williamson nanofluid over a nonlinear stretching plate with activation energy

This study investigates the Williamson nanofluid flow through a nonlinear stretching plate. It aims to analze the global influence of Williamson parameter (λ) rather than local, which is researched for a linear stretching case in the literature. In addition, the features of activation energy are also taken into account in the current review. The developed model with the consequent similarity transformation has still not been perceived. The transformed partial differential equations are solved analytically. The consequences of embedded parameters on the velocity, temperature, and concentration profiles are displayed through figures. Also, the consequences of embedded parameters on skin friction, heat transfer, and mass transfer are demonstrated through tables.     

Investigation of Oldroyd-B fluid flow and heat transfer over a stretching sheet with nonlinear radiation and heat source

The given investigation concerns the study of non-Newtonian Oldroyd-B fluid flow across a permeable surface along with nonlinear thermal radiation, chemical reactions, and heat sources. Equations modified are thus numerically evaluated by employing bvp4c-technique. Obtained outcomes are exhibited graphically. Pictorial notations are used to investigate the consequences of necessary parameters of velocity, energy, and mass. Acquired outcomes provide promising agreement with already established consequences provided in the open literature. The obtained results guided that magnetic field parameter ($M$), porosity parameter ($Kp$), Deborah number ${\beta }_1$ reduce momentum boundary layer thickness, furthermore, growth in the relevant Deborah number ${\beta }_2$ improves the corresponding momentum boundary layer.     

Hall and ion-slip effects on mixed convection flow of Williamson nanofluid over a nonlinear porous stretching sheet with variable thermal conductivity

The intent of this investigation is to analyze the Williamson nanofluid stream past a nonlinearly broadening surface through a leaky medium in the existence of mixed convection, Hall, ion-slip, thermal radiation, and viscous dissipation impacts. Suitable similitude changes give joined nonlinear differential schemes, which were numerically explained via spectral relaxation method. Effectiveness of various physical parameters on velocity ingredients, temperature, and nanoparticle concentration distributions alongside the physical quantities of interests was uncovered graphically. It is found that both velocity profiles increment with an expansion in the Hall parameter. Also, the opposite behavior is noticed for the primary and auxiliary velocity profiles as the ion-slip parameter rises. Moreover, it is observed that the primary velocity and concentration profiles expand with an expansion in the velocity power index parameter, however, the secondary velocity profile reduces. Further, it was showed that the fluid velocities decay while temperature distribution advances by the superior values of the Williamson fluid parameter. Finally, the authenticity of the outcomes was confirmed by contrasting them with prior outcomes under some limited presumptions and discovered to be in terrific understanding.     

Flow of three-dimensional radiative Williamson fluid over an inclined stretching sheet with Hall current and nth-order chemical reaction

In the current communication, three-dimensional Williamson fluid flow past a bidirectional inclined stretching plate with novel Hall current, nonuniform heat source/sink, and nth-order chemical reaction features are investigated. Rosseland's diffusion model is defined for the radiation heat transfer. The nonlinear governing derivative equations satisfying the flow are transmuted to the coupled derivative equations by employing the local similarity quantities and then solved numerically through the Runge–Kutta–Fehlberg method utilizing the shooting quadrature. An inclusive analysis is reported via graphs for the flow rate field, temperature, and concentration distributions for different evolving terms of immense concern. Wall dragging effect and wall heat gradient and wall concentration gradient have been examined, plotted, and described. The detailed geometry reveals that dimensionless velocity field is monotonically rising as the Hall parameter rises. The chemical reaction concentration for the Williamson fluid is enhanced with expanding values of the magnetic field parameter. Transitional values of wall stress components upturn with an increase in Hall parameter while the Williamson term is boosted. Nusselt number is reduced as the Williamson term rises and the Sherwood number enhances with a rising chemical reaction term. The results are verified for limiting cases by comparing with various investigations and found to have excellent accuracy.     

Mixed convection of transient MHD stagnation point flow over a stretching sheet with quadratic convection and thermal radiation

This study investigates the influence of quadratic (nonlinear) convection in transient magnetohydrodynamic (MHD) combined convection over a two-dimensional stretching sheet. It explores the effects of thermal radiation, suction, and heat sources or sinks. The Crank–Nicholson implicit finite difference method is employed for numerical computations. The significance lies in considering secondary convection, which is crucial in understanding nonlinear convective effects affecting flow and heat transfer properties. The study aims to advance our comprehension of how secondary convection impacts the overall system behavior. Through numerical calculations validated against existing literature, strong agreement is demonstrated. The study evaluates secondary convection effects, magnetic, buoyancy, gravitational parameters, Prandtl number, and radiation parameters. Notably, strong quadratic convection alters flow patterns, affecting velocity and temperature profiles. Moreover, it is observed that when increasing the nonlinear (quadratic) convective factors, the temperature profile increases for $$ and decreases for $$. The prevalence of nonlinear or second-order convection highlights magnetic dominance. In essence, this research enhances our understanding of complex convection interactions in MHD flow, shedding light on the role of secondary convection in shaping system behavior.     

Forced convection heat transfer of Williamson fluid flow in porous media over horizontal plate with constant heat flux

Forced convection of Williamson fluid flow in porous media under constant surface heat flux conditions is investigated numerically. A model of Darcy–Forchheimer–Brinkman is used and the corresponding governing equations are expressed in dimensionless forms and solved numerically using bvp4c with MATLAB package. Boundary layer velocity, shear stress, and temperature profiles, in addition to the local Nusselt number parameter over a horizontal plate, are found. The effects of the Forchheimer parameter, Nusselt number, Darcy parameter, porous inertia, and Williamson parameter on the velocity profiles, temperature profiles, coefficient of friction, and coefficient of heat transfer are investigated. The results showed that as the Darcy parameter increases, boundary layer velocity and shear stress increase, while the temperature and Nusselt number decrease. In addition, as Williamson's parameter increases, velocity within the boundary layer, shear stress, and Nusselt number decrease while the temperature profile increases. Also, with larger values of the Forchheimer parameter, the velocity of the boundary layer, shear stress, temperature, and Nusselt number increase. Furthermore, the Nusselt number and the coefficient of friction are obtained on the surface of the horizontal plate.     

Radiation and Hall effects on a 3D flow of MHD Williamson fluid over a stretchable surface

Magnetohydrodynamics (MHD) three-dimensional flow of an unsteady Williamson fluid on an enlarging surface with Hall current, radiation, heat source/sink, and chemical reaction is investigated in this article. The basic governing equations are transformed into a system of ordinary differential equations by using an appropriate similarity transformation. The system is deciphered using the shooting method. The properties of influential parameters such as parameters of magnetic field, Hall current, radiation, and so forth, on the flow are discussed with the help of graphs and tables. We noticed that the increase in the magnetic field reduces the velocity in x-direction and the rate of heat and mass transfer. We also acknowledged that the growing values of Hall current parameter boost the velocity in z-direction but it reduce the temperature and concentration distributions, respectively. The results of this study represent many applications in biomedical engineering and these results are helpful for further study of non-Newtonian fluids in various circumstances.     

Thermophoresis and Brownian motion effects on Williamson nanofluid flow past a stretching surface with thermal radiation and chemical reaction

The heat transfer mechanism of nanofluids has numerous industrial applications owing to the non-Newtonian behavior and has been exercised as a thermophysical phenomena in presence of thermal radiation. The present paper deals with the thermal transfer characteristics of time-independent magnetohydrodynamics Williamson fluid past a stretching surface in presence of the reaction of chemical equilibrium is dealt. The flow constitutive nonlinear partial differential coupled equations are transmitted into ordinary differential equalities by employing relevant similarity transmutations. These deduced equations are determined by using the Runge–Kutta numerical technique with a shooting approach with the aid of MATLAB software. Influences of distinct pertinent flow parameters like an inclined uniform magnetic field, Soret number, heat generation/absorption, and Schmidt number constrained to convective boundary condition is displayed through graphs with relevant physical interpretations. Computed numerical values for the friction factor coefficient, local Nusselt parameter, and Sherwood number are tabulated.      

Fluid properties of an unsteady MHD free stream flow over a stretching sheet in presence of radiation

Variable fluid properties with thermal radiation in an unsteady magnetohydrodynamics free stream incompressible flow over a stretching sheet has been considered. The thermal diffusivity and viscosity of the fluid varies linearly with temperature. The governing partial differential equations are moulded to ordinary differential equations using time-dependent similarity variables and the stream function. RKF technique with shooting method has been implement to find the solution numerically. In the current analysis the impact of unsteadiness, magnetic field, radiative parameter, variable fluid viscosity and thermal diffusivity parameter on heat and flow behavior with the free stream parameter have been studied. Transition point observed in the velocity profiles with an change in unsteadiness parameter and the effect of magnetic field is reduced in the presence of free stram velocity. The velocity and the temperature gradient are computed on the surface and their outcomes with different parameters have been analyzed in the results shown graphically and in tabular form.     

Free convective magnetohydrodynamic flow over an exponentially stretching sheet with radiation

The study explores a steady two‐dimensional magnetohydrodynamic boundary layer flow phenomenon of an incompressible viscous fluid with buoyancy‐driven force over an exponentially stretching sheet. In addition to that, the interaction of thermal radiation in conjunction with dissipative effects, that is, viscous and Joule dissipation is also considered, which is justified due to the presence of magnetic field. The boundary layer equations governed by the flow phenomena are transformed into ordinary differential equations by a suitable choice of similarity transformation. Numerical methods, such as fourth‐order Runge‐Kutta scheme in association with the shooting technique is employed to get an approximate solution of these transformed equations. The numerical computations for the wall shear stress and the heat transfer coefficients are obtained, analyzed, and then discussed. Furthermore, the major findings are pick‐in velocity distribution near the plate is marked with an increase in buoyancy parameter and the rate of heat transfer profile is linear in its boundary layer for low Prandtl number.     

Dual solution of heat transfer in hydromagnetic micropolar fluid flow over a stretching sheet in a porous media: A numerical approach

In this study, the researcher looks at the heat transmission of an incompressible magnetohydrodynamics micropolar fluid across a moving stretched surface in a Darcian permeable medium. The proper boundary conditions are used to facilitate the numerical solution (bvp4c) of the transformed governing equations. Graphical discussions have been made of the influence of the physical parameters on the velocity, angular velocity (microrotation), and temperature, and the distributions are accentuated on the plots via MATLAB. The study is validated by the previous work and it is found appropriate for investigation, where the absolute difference between the previous work and the present investigation by adopting the finite difference scheme is smaller than which implies that the scheme is stable and convergent. The microrotation has a great impact on the micropolar fluid with the influences of buoyancy forces, source, and suction over the stretching surface in a Darcian regime. With a rise in the heat source parameter, both velocity and microrotational profiles lessen, but the opposite is true for temperature. Eringen number () rises with the flow velocity, whereas temperature and microrotational profiles show the reverse relationship. The current study focused on particular applications in non-Newtonian fluid mechanics, polymer flows in filtration systems and metallurgical procedures that included cooling unbroken strips or filaments via a static fluid.     

Symmetry analysis of MHD Casson fluid flow for heat and mass transfer near a stagnation point over a linearly stretching sheet with variable viscosity and thermal conductivity

In this article, the impacts of variable viscosity and thermal conductivity on magnetohydrodynamic, heat transfer, and mass transfer flow of a Casson fluid are analyzed on a linearly stretching sheet inserted in a permeable medium along with heat source/sink and viscous dissipation. To reduce the ascendant partial differential equations into ordinary differential equations, Lie group transformation is utilized. Further, the fourth-order Runge–Kutta strategy is utilized to solve the ordinary differential equations numerically. The numerical results obtained for various parameters by employing coding in MATLAB programming are investigated and considered through graphical representation and tables. We anatomize the impacts of distinctive parameters on velocity, temperature, and concentration distributions.     

Entropy generation on MHD flow of a Casson fluid over a curved stretching surface with exponential space-dependent heat source and nonlinear thermal radiation

The present model concentrates on entropy generation on a steady incompressible flow of a Casson liquid past a permeable stretching curve surface through chemical reaction and magnetic field effects. The exponential space-dependent heat source cum heat and mass convective boundary conditions are accounted for. The resulting nonlinear boundary layer model is simplified by the transformation of similarity. Chebyshev spectral technique is involved for obtaining numerical results of the converted system of the mathematical models. Behavior of the determining thermo-physical parameters on the profiles of velocity, temperature, concentration, skin friction, heat, mass transfer rate, rate of entropy generation, and finally the Bejan number are presented. The major point of the present investigation show that the curvature term weakens the mass transfer profile as the fluid temperature reduces all over the diffusion regime. A decrease in heat generation strengthens the species molecular bond, which prevents free Casson particle diffusion. Furthermore, the mass transfer field diminishes in suction and injection flow medium.     

Entropy optimized MHD fluid flow over a vertical stretching sheet in presence of radiation

The present article describes the influence of radiation on two-dimensional laminar magnetohydrodynamic fluid flow passing over a convective surface. The behavior of the thermal equation is explored through Joule heating, heat generation/absorption, and viscous dissipation. The aim of this study is to examine the physical behavior of the entropy optimization rate. The Cartesian coordinates system is used to model the flow equations. Using similarity variables, a system of partial differential equations is converted into a system of ordinary differential equations. The problem is solved using HAM. The influence of various pertinent parameters on fluid characteristics is graphically explored. Velocity decreases for an increased amount of magnetic parameter, suction parameter, and velocity slip parameter, while behaves the opposite for Grashof number. Temperature increases for a large amount for Brinkman number, magnetic parameter, and radiation parameter, while decreases for Prandtl number. Entropy generation rate increases for Brinkman number, magnetic parameter, and temperature difference parameter. Bejan number decreases for Brinkman number while behaves the opposite for magnetic parameter and temperature difference parameter. Skin friction decreases for large values of magnetic parameters while behaving the opposite for a large amount of velocity slip parameter. Nusselt number decreases for a large amount of Brinkman number. For a better understanding of the study, comparison between numerical outcomes of entropy generation rate and Bejan number for different values of Prandtl number has been done through tables. Also, numerical outcomes of skin friction and Nusselt number are discussed using pertinent parameters through tables.     

Effects of thermal spot configurations on the flow through porous media driven by natural and forced convection

A study concerning the flow of a Newtonian fluid through a porous medium for the particular case natural convection is produced by hot and cold spots placed in the solid phase is presented. Results involving the interaction of forced convection with thermal spots are reported to visualize the mechanisms associated with the generation of complex flow patterns in the porous medium. For this purpose the computation of a two-field model is carried out. Two systems are studied: one is a rectangular porous cavity (RPC) of square cross section and the other is an annular porous cavity (APC) comprised by two concentric vertical cylindrical walls. It is shown, in general, that the flow patterns associated with each configuration and intensities of the thermal spots may be qualitatively inferred by following rules that are established through a basic study of mixed convection in the RPC.